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A note on biorthogonal ensembles
Mathematical Physics · Submitted 23 Aug 2006
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim We show that the kernel is itself an average of a single ratio of characteristic polynomials.
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The abstract, sentence by sentence
We consider ensembles of random matrices, known as biorthogonal ensembles, whose eigenvalue probability density function can be written as a product of two determinants. These systems are closely related to multiple orthogonal functions. It is known that the eigenvalue correlation functions of such ensembles can be written as a determinant of a kernel function. We show that the kernel is itself an average of a single ratio of characteristic polynomials. In the same vein, we prove that the type I multiple polynomials can be expressed as an average of the inverse of a characteristic polynomial. We finally introduce a new biorthogonal matrix ensemble, namely the chiral unitary perturbed by a source term.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We consider ensembles of random matrices, known as biorthogonal ensembles, whose eigenvalue probability density function can be written as a product of two determinants. | background 0.95 | 0.20 | 0.03 | asserted 0.98 |
| s1 | These systems are closely related to multiple orthogonal functions. | background 1.00 | 0.05 | 0.02 | asserted 0.99 |
| s2 | It is known that the eigenvalue correlation functions of such ensembles can be written as a determinant of a kernel function. | background 1.00 | 0.04 | 0.04 | asserted 1.00 |
| s3 | We show that the kernel is itself an average of a single ratio of characteristic polynomials. | result 0.85 | 0.95 | 0.48 | asserted 1.00 |
| s4 | In the same vein, we prove that the type I multiple polynomials can be expressed as an average of the inverse of a characteristic polynomial. | result 0.86 | 0.93 | 0.17 | asserted 1.00 |
| s5 | We finally introduce a new biorthogonal matrix ensemble, namely the chiral unitary perturbed by a source term. | contribution 1.00 | 0.94 | 0.12 | asserted 1.00 |
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Original abstract
We consider ensembles of random matrices, known as biorthogonal ensembles, whose eigenvalue probability density function can be written as a product of two determinants. These systems are closely related to multiple orthogonal functions. It is known that the eigenvalue correlation functions of such ensembles can be written as a determinant of a kernel function. We show that the kernel is itself an average of a single ratio of characteristic polynomials. In the same vein, we prove that the type I multiple polynomials can be expressed as an average of the inverse of a characteristic polynomial. We finally introduce a new biorthogonal matrix ensemble, namely the chiral unitary perturbed by a source term.
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